No, interaction of parts never "ends".
Let me give you a poetical image:
Particles interact to form atoms, atoms interact to form molecules. Molecules interacts to form (neuro)cells and life processe. These interacts to form low level neural processes. These interacts to form higher level process, etc. somewhere in this hierarchy we get protominds that interact to form intention and even higher we get interacting intentions that form meaning etc.
It never ends but at each new level it changes shape and somewhere it may be too complex to follow.
I meant that the definitions such as the mind are the interactions in your argument. The mind does not emerge from the interactions because the interactions are the emergent properties. Isn't that what your argument really boils down to?
Did you even read my post? Its sole purpose was to show the layers of emergence.